Motion of slightly compressible fluids in a bounded domain. II

TL;DR

This paper investigates the behavior of slightly compressible inviscid fluids in bounded domains, establishing existence, uniqueness, and differentiability of solutions, and comparing them to incompressible fluid solutions under small initial divergence.

Contribution

It demonstrates the existence and uniqueness of solutions for compressible fluids and shows their close relation to incompressible solutions when initial divergence is small.

Findings

Solutions depend differentiably on initial data.

Compressible solutions are close to incompressible solutions with similar initial conditions.

Initial divergence of velocity is assumed to be small.

Abstract

We study the problem of inviscid slightly compressible fluids in a bounded domain. We find a unique solution to the initial-boundary value problem and show that it is near the analogous solution for an incompressible fluid provided the initial conditions for the two problems are close. In particular, the divergence of the initial velocity of the compressible flow at time zero is assumed to be small. Furthermore we find that solutions to the compressible motion problem in Lagrangian coordinates depend differentiably on their initial data, an unexpected result for this type of non-linear equations.